304 research outputs found
Fibrational induction meets effects
This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs’ fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type μ(TF) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that μ(TF) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring
New constructions of twistor lifts for harmonic maps
We show that given a harmonic map from a Riemann surface to a
classical compact simply connected inner symmetric space, there is a
-holomorphic twistor lift of (or its negative) if and only if it
is nilconformal. In the case of harmonic maps of finite uniton number, we give
algebraic formulae in terms of holomorphic data which describes their extended
solutions. In particular, this gives explicit formulae for the twistor lifts of
all harmonic maps of finite uniton number from a surface to the above symmetric
spaces.Comment: Some minor changes and a correction of Example 8.
A factorization of a super-conformal map
A super-conformal map and a minimal surface are factored into a product of
two maps by modeling the Euclidean four-space and the complex Euclidean plane
on the set of all quaternions. One of these two maps is a holomorphic map or a
meromorphic map. These conformal maps adopt properties of a holomorphic
function or a meromorphic function. Analogs of the Liouville theorem, the
Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem,
the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and
branch points of a super-conformal map are obtained.Comment: 21 page
Programming Heterogeneous Parallel Machines Using Refactoring and Monte-Carlo Tree Search
Funding: This work was supported by the EU Horizon 2020 project, TeamPlay, Grant Number 779882, and UK EPSRC Discovery, Grant Number EP/P020631/1.This paper presents a new technique for introducing and tuning parallelism for heterogeneous shared-memory systems (comprising a mixture of CPUs and GPUs), using a combination of algorithmic skeletons (such as farms and pipelines), Monte–Carlo tree search for deriving mappings of tasks to available hardware resources, and refactoring tool support for applying the patterns and mappings in an easy and effective way. Using our approach, we demonstrate easily obtainable, significant and scalable speedups on a number of case studies showing speedups of up to 41 over the sequential code on a 24-core machine with one GPU. We also demonstrate that the speedups obtained by mappings derived by the MCTS algorithm are within 5–15% of the best-obtained manual parallelisation.Publisher PDFPeer reviewe
Willmore Surfaces of Constant Moebius Curvature
We study Willmore surfaces of constant Moebius curvature in . It is
proved that such a surface in must be part of a minimal surface in
or the Clifford torus. Another result in this paper is that an isotropic
surface (hence also Willmore) in of constant could only be part of a
complex curve in or the Veronese 2-sphere in . It is
conjectured that they are the only examples possible. The main ingredients of
the proofs are over-determined systems and isoparametric functions.Comment: 16 pages. Mistakes occured in the proof to the main theorem (Thm 3.6)
has been correcte
Localized induction equation and pseudospherical surfaces
We describe a close connection between the localized induction equation
hierarchy of integrable evolution equations on space curves, and surfaces of
constant negative Gauss curvature.Comment: 21 pages, AMSTeX file. To appear in Journal of Physics A:
Mathematical and Genera
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